New York businessman James Lieto was an innocent bystander in a fraud investigation last year. Federal agents seized $392,000 of his cash anyway. An armored-car firm hired by Mr. Lieto to carry money for his check-cashing company got ensnared in the FBI probe. Agents seized about $19 million—including Mr. Lieto's money—from vaults belonging to the armored-car firm's parent company. He is one among thousands of Americans in recent decades who have had a jarring introduction to the federal system of asset seizure. Some 400 federal statutes—a near-doubling, by one count, since the 1990s—empower the government to take assets from convicted...